few_shots: Example:
Problem/activity:
Narrative
The purpose of this activity is for students to create composite shapes and compose new shapes from composite shapes. For example, students may find that a rectangular prism can be made from four triangular prisms or from eight smaller cubes. Students then discover that they can combine two of the composed rectangular prisms to make a new rectangular prism.
Required Materials
Materials to Gather
Geoblocks
Launch
Groups of 4
Give each group geoblocks.
Display a large cube.
“This cube can be built from other shapes. See how many ways you can build a cube using other shapes.”
3 minutes: partner work time
Share responses.
Activity
“Choose a different shape. Try to find smaller shapes you can use to build that shape.”
5 minutes: partner work time
Student Response
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Activity Synthesis
Display a cube and a composite cube—for example, a cube composed of four small cubes and one rectangular prism.
Demonstrate putting the shapes together.
“What new shape did I make with the cube and the cube I made?”
“Put your shapes together to make a rectangular prism.”
Display a rectangular prism and two triangular prisms.
“Find these shapes. Put them together. What new shape can you make?” (a cube, a rectangular block)


Standard options: 
A. Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.
B. Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size); build and draw shapes to possess defining attributes.
C. Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape. (Students do not need to learn formal names such as “right rectangular prism.”)


Thought: This activity asks students to compose three-dimensional shapes, using geoblocks, to create composite shapes.
Answer: C

Now, assign the following problem or activity to one or more relevant standards it aligns with.
shot_info: im_lesson_009431

1
options: Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

Explain a proof of the Pythagorean Theorem and its converse.
problem_activity: Activity
The purpose of this activity is to give students practice finding missing side lengths in a right triangle using the Pythagorean Theorem.
Teacher Notes for IM 6–8 Math Accelerated
Adjust time to 15 minutes. After 2–3 minutes of work time, select 1–2 groups to share their work for the first question before continuing with the remaining questions. If time is limited, tell students that the final question is optional.
Launch
Arrange students in groups of 2. Give students 10 minutes of quiet work time and then have them compare with a partner. If partners disagree about any of their answers, ask them to explain their reasoning to one another until they reach agreement. Follow with a whole-class discussion.
Representation: Internalize Comprehension.
Activate or supply background knowledge by displaying the Pythagorean Theorem with a labeled diagram. Allow students to use calculators to ensure inclusive participation in the activity.
Supports accessibility for: Memory; Conceptual processing
Student Facing
Find
\(c\)
.
[IMG]
Find
\(b\)
.
[IMG]
A right triangle has sides of length 2.4 cm and 6.5 cm. What is the length of the hypotenuse?
A right triangle has a side of length
\(\frac14\)
and a hypotenuse of length
\(\frac13\)
. What is the length of the other side?
Find the value of
\(x\)
in the figure.
[IMG]
Student Response
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click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
The spiral in the figure is made by starting with a right triangle with both legs measuring one unit each. Then a second right triangle is built with one leg measuring one unit, and the other leg being the hypotenuse of the first triangle. A third right triangle is built on the second triangle’s hypotenuse, again with the other leg measuring one unit, and so on.
[IMG]
Find the length,
\(x\)
, of the hypotenuse of the last triangle constructed in the figure.
Student Response
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click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
Ask students to share how they found the missing side lengths. If students drew triangles for the two questions that did not have an image, display a few of these for all to see, noting any differences between them. For example, students may have drawn triangles with different orientations or labeled different sides as
\(a\)
and
\(b\)
.
For the last question, ask students to say what they did first to try and solve for
\(x\)
. For example, while many students may have found the length of the unknown altitude first and then used that value to find
\(x\)
, others may have set up the equation
\(34-5^2=18-x^2\)
.
Point out that when you know two sides of a right triangle, you can always find the third by using the Pythagorean identity
\(a^2 + b^2 = c^2\)
. Remind them that it is important to keep track of which side is the hypotenuse.
Reading, Writing, Speaking: MLR3 Clarify, Critique, Correct.
Before students share their method for the questions that did not have an image, present an incorrect solution based on a common error related to labeling the sides of a right triangle. For the right triangle with a side of length
\(\frac{1}{4}\)
and a hypotenuse of length
\(\frac{1}{3}\)
, draw a right triangle with the legs labeled
\(\frac{1}{4}\)
and
\(\frac{1}{3}\)
. Provide an incorrect explanation such as: “I know that
\(a = \frac{1}{4}\)
and
\(b = \frac{1}{3}\)
, so when I use the Pythagorean Theorem, I get the equation
\((\frac{1}{4})^2 + (\frac{1}{3})^2 = c^2\)
.” Ask students to identify the error, critique the reasoning, and revise the original statement. As students discuss in partners, listen for students who clarify the meaning of the hypotenuse and identify
\(c\)
as the length of the hypotenuse in the Pythagorean Theorem. This routine will engage students in meta-awareness as they critique and correct a common error when labeling the sides of a right triangle.
Design Principles(s): Cultivate conversation; Maximize meta-awareness


correct_option_index: 1